Nonconformal entanglement entropy
Implicitly this expression assumes that
Nonconformal entanglement entropy
Marika Taylor 0 1
William Woodhead 0 1
0 High eld , Southampton, SO17 1BJ , U.K
1 Mathematical Sciences and STAG Research Centre, University of Southampton
We explore the behaviour of renormalized entanglement entropy in a variety of holographic models: nonconformal branes; the Witten model for QCD; UV conformal RG ows driven by explicit and spontaneous symmetry breaking and Schrodinger geometries. Focussing on slab entangling regions, we nd that the renormalized entanglement entropy captures features of the previously de ned entropic cfunction but also captures deep IR behaviour that is not seen by the cfunction. In particular, in theories with symmetry breaking, the renormalized entanglement entropy saturates for large entangling regions to values that are controlled by the symmetry breaking parameters.
AdSCFT Correspondence; Gaugegravity correspondence

4.1
4.2
2 Renormalized entanglement entropy
3 AdS entanglement entropy in general dimensions
4 Nonconformal branes
Entanglement functional and surfaces
Witten model
5 Renormalized entanglement entropy for RG ows
1 Introduction
S = c1 D
+ c0 ln(R= ) + c~0;
{ 1 {
Spontaneous symmetry breaking: Coulomb branch of N = 4 SYM
5.1.1
5.1.2
Coulomb branch disk distribution
Coulomb branch spherical distribution
5.2 Operator driven RG ow
6 Nonrelativistic deformations
7 Interpretation and comparison to QFT results
8 Conclusions and outlook
1
Introduction
Entanglement entropy is widely used in condensed matter physics, quantum information
theory and, more recently, in high energy physics and black holes. Consider a reduced
density matrix
A, obtained from tracing out certain degrees of freedom from a quantum
system. The associated entanglement entropy is then the von Neumann entropy:
S =
Tr ( A ln A) :
Throughout this paper we will be interested in the case for which a quantum system is
subdivided into two, via partitioning space. In such a case A is a spatial region, with
The entanglement entropy characterizes the nature of the quantum state of a system.
For example, in the ground state of a quantum critical system in D spatial dimensions:
where c1 D, c0 and c~0 are dimensionless; R is a characteristic scale of the region A and is
an UV cuto . Logarithmic terms arise when D is odd, and their coe cients are related to
the a anomalies of the stress energy tensor. More generally, the famous area law leading
term characterizes the ground state of a system and can be used to test trial ground state
wavefunctions. Entanglement entropy can also be used to distinguish between di erent
phases of a system, such as the con ning/decon ning phase transition [1].
Continuum quantum
eld theory (with a cuto ) is often used as a tool to describe
discrete condensed matter systems. In this context, the cuto appearing in (1.2) is related
to the underlying physical lattice scale in the discrete system and the coe cients of power
law terms such as c1 D capture the leading physical contributions to the entanglement
eld theory perspective, the expansion in (1.2) implicitly assumes
the use of a direct energy cuto
as a regulator. Di erent methods of regularisation result
in di erent regulated divergences and thus the power law divergences are often called
nonuniversal. By contrast logarithmic divergences are often denoted as universal as their
coe cients are related to the anomalies of the theory.
In even spatial dimensions, the logarithmic term in (1.2) is absent but the constant term
c~0 is believed to be related to the number of degrees of freedom of the system. However,
c~0 is manifestly dependent on the choice of the cuto . In two spatial dimensions, if
(1.3)
(1.4)
(1.5)
(1.6)
for a spatial region with boundary of length R, then changing the cuto as
R
S = c 1
+ c~0
! 0 [2{4]. However, such an approach has several drawbacks. The
regularisation is speci c to the shape of the geometry (a slab) and a modi ed prescription
is needed for curved entangling region boundaries such as spheres, for which the scale of
the entangling region is related to the local curvature of the entangling region boundary
(see proposals in [5]). Any such prescription depends explicitly on the UV behaviour of
the theory. More generally, extraction of nite terms by di erentiation obscures scheme
dependence: there is no connection with the renormalization scheme used for other QFT
quantities such as the partition function and correlation functions.
From a quantum
eld theory perspective, as opposed to a condensed matter
perspective, it is very unnatural to work with a regulated rather than a renormalized quantity. In
previous papers [6, 7], we introduced a systematic renormalization procedure for
entanglement entropy, in which the counterterms are inherited directly from the partition function
counterterms. As we review in section 2, such renormalization guarantees that the
counterterms depend only on the quantum
eld theory sources (nonnormalizable modes in
holographic gravity realisations) and not on the state of the quantum eld theory
(normalizable modes in holographic gravity realisations).
The renormalized entanglement entropy Sren expressed as a function of a characteristic
scale of the entangling region implicitly captures the behaviour of the theory under an RG
ow: small entangling regions probe the UV of the theory, while larger regions probe the
IR. In this paper we will establish how these
nite contributions to entanglement entropy
behave in a variety of theories, using holographic models.
The plan of this paper is as follows. In section 2 we review the de nition of
renormalized entanglement entropy introduced in [6]. In section 3 we calculate the renormalized
entanglement entropy for a slab region in antide Sitter (in general dimensions). The latter
is relevant for the nonconformal branes discussed in section 4, as the latter can be viewed
as dimensional reductions of antide Sitter theories in general dimensions. In section 4
we also compute the renormalized entanglement entropy for a slab region in the Witten
holographic model for QCD. Section 5 explores renormalized entanglement entropy for
operator and driven holographic RG
ows, which are UV conformal. In section 6 we consider
renormalized entanglement entropy in holographic Schrodinger geometries. In section 7
we summarise the main features of the renormalized entanglement entropy, using both our
holographic results and earlier perturbative/lattice calculations. We conclude in section 8.
2
Renormalized entanglement entropy
Entanglement entropy is usually calculated using the replica trick. The Renyi entropies
where Z(1) is the partition function and Z(n) is the partition function on the replica space
obtained by gluing n copies of the geometry together along the boundary of the entangling
region. The entanglement entropy is obtained as the limit
Note that this limit implicitly assumes that the Renyi entropies are analytic in n.
Both sides of (2.1) are UV divergent. In a local quantum
eld theory, the UV
divergences of log Z(n) cancel with those of n log Z(1) except at the boundary of the entangling
region; therefore the U V divergences of S(n) scale with the area of this boundary.
S = nli!m1 Sn:
{ 3 {
(2.1)
(2.2)
We can formally de ne the renormalized entanglement entropy as [6]
Snren =
1
(1
n)
(log Zren(n)
n log Zren(1))
(2.3)
with Sren = S1ren. Here the renormalized partition functions are de ned with any suitable
choice of renormalization scheme.
The replica space matches the original space, except at the boundary of the entangling
region where there is a conical singularity. To de ne the renormalization on the replica
space it is therefore natural to work within a renormalization method that works for generic
curvature backgrounds for the quantum eld theory.
is the UV cuto , Vd is the volume of the background (Euclidean) geometry, m2 is
the mass and R is the Ricci scalar. The coe cients (ad; ad 2; bd 2;
and in the above expressions we ignore boundaries of M.
) are dimensionless
The divergences of the partition function on the replica space Z(n) have exactly the
same structure and coe cients. However, the curvature of the replica space has an
additional term from the conical singularity [8, 9]
1)2;
where (@ ) is localised on a constant time hypersurface, on the boundary of the
entangling region. (Here and in what follows we consider only static situations.) Therefore,
when we use the replica formula (2.2) the leading divergences of the partition functions
(scaling with the volume) cancel so that the leading divergent term in the entanglement
entropy behaves as
Sreg = 4 bd 2 d 2
d
d 2xp
+
Such a divergence can clearly be cancelled by the counterterm
Sct =
4 bd 2 d 2
d
d 2xp ;
which is covariantly expressed in terms of the geometry of the entangling region.
Z
2.2
In gaugegravity duality, the de ning relation is [
10, 11
]
IE =
log Z;
(2.8)
where IE is the onshell action for the bulk theory dual to the eld theory. In the
supergravity limit this is given by the onshell Euclidean EinsteinHilbert action together with
appropriate matter terms i.e.
IE =
1
Z
d p
d x h (K +
) ;
(2.9)
where the latter is the usual GibbonsHawkingYork boundary term. The volume
divergences of the bulk gravity action correspond to UV divergences of the dual quantum
eld
theory; these divergences can be removed by appropriately covariant counterterms at the
conformal boundary.
gravity the action counterterms are
For example, in the case of asymptotically locally antide Sitter solutions of Einstein
Ict =
1
Z
d p
d x h (d
1) +
R
2(d
2)
+
where the ellipses denote terms of higher order in the curvature and logarithmic
counterterms arise for d even.
Applying the replica formula to the bulk terms in the action, as discussed in [12], and
using the analogue of (2.5) for the bulk curvature, namely,
(2.10)
(2.11)
(2.12)
(2.13)
gives the RyuTakayanagi functional [13] for the entanglement functional:
Applying the replica formula to the counterterms gives
p (1 +
) ;
with the leading counterterm being proportional to the regulated area of the entangling
surface boundary. Analogous expressions for higher derivative gravity and gravity coupled
to scalars can be found in [6].
Using a radial cuto
to regulate is perhaps the most geometrically natural way to
renormalize the area of the minimal surface but it is not the only holographic
renormalization scheme. Dimensional renormalization for holography was developed in [14] and this
method could also be used to renormalize the holographic entanglement entropy.
{ 5 {
AdS entanglement entropy in general dimensions
In this section we review the renormalized entanglement entropy for a slab domain in
Antide Sitter in general dimensions. The regulated entanglement entropy for such slab domains
was analysed in [13]; here we extract from their analysis the renormalized entanglement
entropy, in general dimensions. This quantity is relevant to the nonconformal brane
backgrounds discussed in the next section, as the latter can be understood in terms of parent
Antide Sitter theories, and also relevant for the Schrodinger backgrounds discussed in
section 6.
Let us parameterise AdSD+2 as
counterterm is the regulated area of the boundary i.e.
Z
d
D 1p~
h
The entangling functional is
We now consider an entangling region in the boundary of width L in the x direction, on
a constant time hypersurface, longitudinal to the other (D
1) coordinates y . The bulk entangling surface is then speci ed by the hypersurface x( ) minimising
where x0 = @ x. The equation of motion admits the rst integral
where 0 is the turning point of the surface, related to L via
or equivalently
L = 2 0
xDdx
x2D
The regulated onshell value of the entangling functional is then
Sreg =
Vy
2GD+2
Dq
d
1
2D
Dd
q 2D
0
= 0
(where we assume that D 6= 1) and therefore
which can be rewritten in terms of dimensionless quantities as
Dq
d
1
2D
2D
0
(D
(D
1
2
3
1
1) D 1 5 ;
1
1)~D 1 :
This can be evaluated to give and hence
As we discuss later, this quantity is closely related to the entropic c function for slabs in
antide Sitter computed in [3]. In the case of D = 1 (AdS3) the entangling functional is
logarithmically divergent, and the renormalized entanglement entropy depends explicitly
on the renormalization scale: for a single interval
Z
d10xpGN e
2
2(8
1
p)!N 2 jF8 pj
2
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(4.1)
(4.2)
HJEP01(28)4
where the constants ( ; ) are given below for Dpbranes and fundamental strings
respectively. (Note that it is convenient to express the eld strength magnetically, so for p < 3
we use Fp+2 =
equations following from the action above can be reduced over a sphere, truncating to a
(p + 2)dimensional metric and scalar. The resulting action is then
Id+1 =
N
Z
dd+1xpge
{ 7 {
where
coupling as
p =
2(p 4)
2 p 5 (5
9 p p+1
p) p 5 p 5
and gd2 is the dimensionful coupling of the dual eld theory, which is related to the string
At any length scale l there is an e ective dimensionless coupling constant
where d = p + 1 and the constants (N ; ; ; C) depend on the type of brane under
consideration.
For Dpbranes =
2 d 2 + dx dxd
=
(p
7)(p
3)
4(p
5)
{ 8 {
For the fundamental string = 2 3
N =
3
gsN 2 ( 0)1=2
p
and the dimensionful coupling is
so
AdSd+1 solution:
In all cases, the dual frame is chosen such that the equations of motion admit an
where the constant
again depends on the case of interest: for Dpbranes
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
HJEP01(28)4
=
For further discussion of this point, see [16].
Let us de ne a parameter as The nonconformal branes are formally related to AdS gravity in the following way [17].
HJEP01(28)4
Now we consider (2 + 1)dimensional gravity with cosmological constant
=
(2
1),
I(2 +1) =
NAdS
Z
d
2 +1xpg2 +1 (R2 +1 + 2 (2
1)) :
so that the action is
Reducing on a (2
ansatz
results in the action (4.2) where
d)dimensional torus with coordinates za via a diagonal reduction
ds2 = ds2d+1(x) + exp
(2
2
d)
dzadza
while
=
3=4 for fundamental strings. In general the equations admit an AdS solution
with linear dilaton provided that the parameters are related as
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
N = NAdSV(2 d);
with V(2 d) the volume of the compacti cation torus.
4.1
Entanglement functional and surfaces
The entanglement functional follows from the replica trick: in the dual frame
The equations for the entangling surface can be expressed geometrically as
S = 4 N
Z
d
d 1 p
x he
Km =
where gmn is the background metric, hij is the induced metric on the entangling surface,
Xm(xi) speci es the embedding of the entangling surface into the background and Km are
the associated traces of the extrinsic curvatures.
The dual frame entanglement functional follows directly from the reduction of the pure
gravity entanglement functional
S = 4 NAdS
d
The renormalized entanglement entropy for a strip in the F1 background can be
exwhen one again uses the diagonal reduction ansatz (4.15), and assumes that the entangling
surface wraps the torus and that the shape of the surface does not vary along the torus
directions. In the upstairs picture the entangling surface satis es
KM = 0;
where the background metric is now denoted g(2 +1)MN and KM denotes the traces of
the extrinsic curvatures. Thus, any AdS entangling surface which factorises as 2 1 =
T (2
d)
will give an entangling surface for nonconformal branes; moreover, the
nonconformal brane surface will inherit its renormalized entanglement entropy from the
upstairs entangling surface.
entangling functional is
As an example, let us consider slab entangling regions, characterised by a width
x =
L. The bulk entangling surface is speci ed as x( ) and in the background (4.10) the
S = 4 N Vy
Z
d
which is indeed precisely the functional obtained in (3.3), identifying D = (2
renormalized entanglement entropy can then be expressed as
pressed as
where
(4.20)
(4.21)
(4.22)
(4.23)
(4.24)
(4.25)
(4.26)
where the e ective coupling is expressed as ge2 (L) = gf2N L2. The expression for the
renormalized entanglement entropy of a strip in the D1 background is analogous:
4.2
Witten model
sixdimensional background:
The Witten [18] holographic model for YM4 can be expressed in terms of the following
Sren =
Sren =
Regularity of the geometry requires that the circle direction
must have periodicity
2
3
L =
KK :
This model originates from D4branes wrapping the circle
with antiperiodic boundary
conditions for the fermions. which breaks the supersymmetry. At low energies the model
resembles a fourdimensional gauge theory, with the gauge coupling being g2 = g52=L . The
gravity solution captures the behaviour of this theory in the limit of large 't Hooft coupling
2 = g2N
and Sugimoto [19, 20] introduced D8branes wrapped around the S4 on which the theory
is reduced from ten to six dimensions. These D8branes model chiral avours in the dual
gauge theory and the resulting WittenSakaiSugimoto model has been used extensively as
a simple holographic model of a nonsupersymmetric gauge theory with avours.
The operator content of the dual theory captured by the metric and scalar eld is
the fourdimensional stress energy tensor Tab, a scalar operator O
component of the
vedimensional stress energy tensor T
corresponding to the
and the gluon operator O
corresponding to the bulk scalar eld. These operators satisfy a Ward identity [16]
One of the main applications of this model is in the context of avour physics: Sakai
and their expectation values can be extracted from the above geometry. For example, the
condensate of the gluon operator
hTaai + hO i +
1
g2 hOi = 0
hOi =
2
5 2 2N
37
L4
and therefore L controls the QCD scale of the theory.
Next we can consider a slab entangling region, wrapping the circle direction ,
characterised by a width
x = L. Entanglement entropy in this theory was previously discussed
in [1], with the con nement transition being associated with a discontinuity in the
derivative of the entanglement entropy with respect to L. The bare entanglement functional is
S = 4 N V2L
5
Z d p1 + f ( )(x0)2
where V2 is the volume of the twodimensional crosssection of the slab. The entanglement
can then be written as
Sreg = 8 N V2L
Z
d
pf ( ) 10
5 pf ( ) 10
f ( ) 10
where
is the turning point of the surface, related to the width of the entangling region as
L = 2
Z
r
f ( )
d
10f( )
10f( )
1
:
0.0
0.1
solid orange lines indicate the renormalized entropy for the two possible connected minimal surfaces.
The entanglement entropy can be renormalized as before, with the counterterm
contributions being
Sct =
2 N V2 4
:
L
For large entangling regions, the only possible entangling surface is the disconnected
conguration, for which the renormalized entanglement entropy is
Sren = 8 N V2L
=
For small entangling regions the condensate is negligible and the renormalized entanglement
entropy is controlled by the conformal structure
The renormalized entanglement entropy is plotted in gure 1. As discussed in [1] there
is a discontinuity in the derivative of the entanglement entropy for slab widths around
L
0:4 KK . For larger values of L the entanglement entropy saturates at a constant value.
Renormalized entanglement entropy for RG ows
In this section we will consider holographic entanglement entropy in geometries dual to RG
ows. We work in Euclidean signature with a bulk action
I =
1
1
2
Holographic RG ows with at radial slices can be expressed as
ds2 = dr2 + exp(2A(r))dxidxi;
where the warp factor A(r) is related to a radial scalar eld pro le (r) via the equations
of motion
d
2
dr2 + d
dA
dr
=
dV
d
d2A
dr2 =
1
d
2(d
1) dr
2
:
These equations can always be expressed as rst order equations [21]
where the (fake) superpotential W ( ) is related to the potential as
d
dr
dA
dr
d
dr
= W
=
2(d
1)
dW
d
V =
(d
1)2
2
dW
d
2
d
d
1
!
W 2 :
S =
Vy Z
4G
dre(D 1)A(r)q
1 + e2A(r)(x0)2;
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
(5.8)
(5.9)
Near the conformal boundary the potential can be expanded in powers of the scalar eld as
and hence the superpotential can be written as
V = d(d
1)
= d=2 + pd2 + 4m2=2. The higher order terms in the superpotential are not
unique, as di erent choices are associated with di erent RG
ows.
Note that for at domain walls, a single counterterm (in addition to the usual
Gibbons
Hawking term) is su cient
Ict =
(d
1) Z
although the derivation of the entanglement entropy counterterms requires knowledge of
the counterterms for a curved background (since the replica space is curved).
The entanglement entropy for a slab region
x = L in the RG
ow geometry is
where D is the number of spatial directions in the dual theory, Vy is the regulated volume
of the longitudinal directions and x(r) de nes the entangling surface. Then
where at the turning point r0 of the surface A(r) = A0. The regulated onshell action is
with the cuto
being r = .
The entanglement entropy counterterms for RG ows driven by relevant deformations
were discussed in [6], working perturbatively in the deformation. Here we will analyse both
spontaneous and explicit symmetry breaking, using exact supergravity solutions.
Spontaneous symmetry breaking: Coulomb branch of N = 4 SYM
In this section we consider the case of VEV driven ow, i.e. spontaneous symmetry breaking.
In such a situation, the scalar eld has only normalizable modes and thus asymptotically
the scalar eld behaves as
where (0) is related to the operator expectation value as
L = 2
Z 1
eDA0 dr
r0 eA(r)pe2DA(r)
e2DA0
dr
e(2D 1)A(r)
pe2DA(r)
e2DA0
e(d 2) !
(d
2)
:
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
(5.15)
(5.16)
(5.17)
From (5.4) and (5.7), one can immediately read o the asymptotic form of the warp factor:
Substituting into the regulated action, we then obtain
2)=2, and is logarithmically divergent for
= (d
2)=2. (The latter case does not however arise holographically, as when the lower
bound on the conformal dimension is saturated the operator automatically obeys free eld
equations.) Therefore, for VEV driven ows the only counterterm required is the regulated
area of the boundary of the entangling surface:
Sct =
1
Z
4(d
d 2xph:
Note that one can derive the same result from the bulk action counterterms, using the
replica trick; see below for the case of the Coulomb branch of N = 4 SYM. Thus the
renormalized entanglement entropy for slabs in VEV driven ows is
Now let us consider the general structure of the renormalized entropy. In the vacuum of
the conformal eld theory, the renormalized entropy must behave as
with c0 a dimensionless constant on dimensional grounds: the entropy scales with the
longitudinal volume Vy and the width of the entangling region L is the only other dimensionful
scale in the problem. The value of c0 in holographic theories is given in (3.12).
Now working perturbatively in the operator expectation value hOi the renormalized
entropy must behave as
where c1 is dimensionless and we work in a limit in which
(5.18)
(5.19)
(5.20)
(5.21)
(5.22)
(5.23)
(5.24)
(5.25)
i.e. the width of the entangling region is much smaller than the length scale set by the
condensate.
5.1.1
Coulomb branch disk distribution
We now analyse a speci c example: the renormalized entanglement entropy of slab domains
on the Coulomb branch of N = 4 SYM.
We consider the case of a disk distribution of branes preserving SO(4)
SO(2) symmetry, for which the equations of motion follow from (5.1), with the superpotential being [22] Using the standard Fe ermanGraham coordinates near the conformal boundary: The metric in vedimensional gauged supergravity is then
p
6
1
3
4
p
6
ds2 =
2w2
dw2
Here the coordinate w ! 1 at the conformal boundary and
characterises the expectation
value of the dual scalar operator. The scalar eld can be expressed by the relation
= p
2 2 +
1
2
1
6
dx dx
operator, following [23, 24]:
where we use the standard relation between the Newton constant and the rank of the dual
The vanishing of the dual stress energy tensor is required given the supersymmetry but
careful holographic renormalization is required to derive this answer.
The regulated entanglement entropy of a slab domain in this geometry can be written as
HJEP01(28)4
Using the rst integral of the equations of motion the width of the entangling region can
be expressed in terms of the turning point of the surface w0 as
Sct =
V2
8G5
(5.26)
(5.27)
(5.28)
(5.29)
(5.30)
(5.31)
(5.32)
(5.33)
(5.34)
where c is an integration constant and w0 satis es
The regulated entanglement entropy is then
L = 2
cdw
w0 w2 3p 6w6
c2
and the required counterterm is expressed in terms of the regulated area of the boundary
of the entangling surface i.e. there are counterterm contributions
at each side of the slab. (The total contribution is therefore twice this value.) Note that
the counterterms in this case clearly contribute both divergent and nite parts: expanding
in powers of the cuto
It is then convenient to write the entanglement entropy in terms of dimensionless quantities
as
lim
!1
Z ~
y0
p
y y + 1
dy py2(y + 1)
and y0 is the turning point of the surface. Then
L = y0py0 + 1
y0 yp(y + 1)py2(y + 1)
These integrals can be computed numerically. There is a maximal value of L (for xed )
for which a connected entangling surface exists: the critical value of L is such that
1
3
:
2 e p
2
6
1
e
p
6
=
2w2:
The metric in vedimensional gauged supergravity is then
with
2
3
2
p
6
p
6
ds2 =
2w2
dw2
Coulomb branch spherical distribution
We now consider the renormalized entanglement entropy of slab domains on the
Coulomb branch of N = 4 SYM for the case of a spherical distribution of branes, preserving
SO(4)
SO(2) symmetry. The equations of motion follow from (5.1), with the
superpotential being [22]
Here the coordinate w ! 1 at the conformal boundary and
characterises the expectation
value of the dual scalar operator. The scalar eld can be expressed by the relation
Lcrit
1:5708:
Sren =
12G5
:
For L > Lcrit there is no connected entangling surface but the disconnected entangling
surface consisting of two components x =
L=2 and x = L=2 still exists. For the latter
one can straightforwardly calculate the renormalized entanglement entropy as
HJEP01(28)4
The renormalized entanglement entropy is plotted in gure 2: its rst derivative is
discontinuous at L = Lcrit. For small values of L, the analytic expressions (5.19) is valid:
Sren =
V2
G5
0
3
2 !2
1
( 1 )
6
and the constant C1 can be determined as:
C1
0:03137:
(5.35)
(5.36)
(5.37)
(5.38)
(5.39)
(5.40)
(5.41)
(5.42)
(5.43)
2
4
6
8
10
the numerical results for a cuto
of
= 1010, the dotted red line shows the small
L
t of
equation (5.38), the dashed yellow line shows the value of the renormalized entropy for disconnected
surfaces.
Using the standard Fe ermanGraham coordinates near the conformal boundary:
ds2 =
d 2 +
= p
2 2 +
1
2
1
6
1
2
1
18
We can then read o the expectation values of the dual stress energy tensor and scalar
operator, following [23, 24]:
=
where we use the standard relation between the Newton constant and the rank of the dual
The vanishing of the dual stress energy tensor is required given the supersymmetry but
again careful holographic renormalization is required to derive this answer.
The regulated entanglement entropy is then
Sreg =
V2 Z
and the required counterterm is expressed in terms of the regulated area of the boundary
of the entangling surface i.e. there are counterterm contributions
Sct =
V2
8G5
(5.44)
(5.45)
(5.46)
(5.47)
(5.48)
the counterterms in this case clearly contribute both divergent and nite parts: expanding
Sct =
It is then convenient to write the entanglement entropy in terms of dimensionless quantities
as
where ~ is a rescaled dimensionless cuto . Implicitly this expression assumes that 2 6= 0
and y0 is the turning point of the surface. Then
L = y0py0
1
Z 1
dy
y0 yp(y
1)py2(y
1)
y02(y0
1)
These integrals can again be computed numerically. As in the previous case, for xed
there is a maximal value of L for which a connected entangling surface exists. The critical
value is
For lengths grater than the critical length, the minimal surface is disconnected and the
renormalized entanglement entropy can be calculated analytically to give
Lcrit
0:8317
Sren =
V2 2
6G5
:
Lc
0:75
(5.49)
(5.50)
(5.51)
(5.52)
(5.53)
(5.54)
(5.55)
(5.56)
For subcritical values, there are two possible surfaces with turning points y0 for each width
L and one must choose the surface for which the renormalized area is minimised.
The renormalized entanglement entropy is plotted in gure 3. There is a phase
transition between the connected and disconnected entangling surfaces at Lc such that
i.e. Lc < Lcrit, and the entanglement entropy is saturated for L
Lc. In the regime of
small L the analytic expressions (5.19) are valid:
where the constant C1 can be determined as:
C1
0:03167
0.5
1.0
1.5
2.0
2.5
solid orange lines indicate the renormalized entanglement entropy for the two possible connected
minimal surfaces. The dashed orange line indicates the entanglement entropy for the disconnected
surface. The dotted red line shows the small L t of equation (5.55).
In this section we consider the case of an operator driven RG
ow, the GPPZ ow [25].
The equations of motion again follow from (5.1), with the superpotential being
The metric can be expressed as while the scalar eld is given by 1 2
p
3
ds2 =
d 2
2 +
1
2 (1
2 2)dx dx
The scalar is dual to a dimension three operator. By expanding near the conformal boundary and using the holographic renormalization dictionary, [23, 24] showed that the
GPPZ solution is dual to a deformation (proportional to ) of N = 4 SYM by the dimension
three scalar operator, with the expectation values of the operators being
The vanishing stress energy tensor is again required by supersymmetry while the vanishing
of the scalar VEV re ects the explicit (as opposed to spontaneous) symmetry breaking.
=
p
2
3
log
1 +
1
hTij i = hOi = 0:
(5.57)
(5.58)
(5.59)
(5.60)
Now let us consider the renormalized entanglement entropy of a strip region in this
geometry. The entanglement entropy can be expressed as
The overall dependence on the deformation
can be scaled out to give
S =
V2 Z
2G5
d
(1
S =
V2 2 Z
2G5
dv
(1
3
v3
where v =
and X =
x. Then the entangling surface of width L satis es
HJEP01(28)4
L = 2
Z v0
0
dv q
v
3
(1
v2)4
v6 2(1
v2)
where the integration constant
is related to the turning point of the surface v0 by
(5.61)
(5.62)
(5.63)
(5.64)
(5.65)
(5.66)
(5.67)
(5.68)
(5.69)
=
(1
v2)3=2
0
v
3
0
:
Lcrit
0:3008:
As in the previous cases there is a phase transition between a connected solution for L <
Lcrit and a disconnected solution for L >
Lcrit where
The regulated entanglement entropy is then
Sreg =
V2 2 Z v0
2G5
dv
(1
q
v3 (1
v2)5=2
v2)3
v6 2
:
Sct =
d x
2 p~
h 1 +
3
2 2 log( )
The counterterms for the entanglement entropy can be derived from the bulk action
counterterms using the replica trick:
where the cuto in the
coordinates is ~ =
= . Evaluating this counterterm gives a
contribution from each endpoint of the strip:
Sct =
V2
8G5
1
2
1 + 2 log
;
which indeed matches the regulated divergences of (5.66). Thus the total renormalized
entropy is
Sren =
V2 2
0
Z v0
dv
(1
q
v3 (1
v2)5=2
v2)3
v6 2
1
2 2 +
1
2
1
log A :
0.10
0.15
0.5
1.0
1.5
2.0
2.5
ow. The solid blue line and solid orange
lines indicate the renormalized entropy for the two possible connected minimal surfaces. Note that
Sren > 0 near Lcrit for the connected solutions.
These integrals can once again be evaluated numerically, the results of which are plotted
in gure 4. As in the case of the spherical brane distribution, there are two possible turning
points for a given length L < Lcrit, the branch with v0 < v0;crit is favoured for all such L.
Both branches are positive near L = Lcrit, whereas it can be shown analytically that the
renormalized entropy is zero for disconnected surfaces and so there is a transition from the
connected to disconnected surface solutions at around
Lc
0:27
(5.70)
where the entanglement entropy has a discontinuous derivative.
6
Nonrelativistic deformations
Entanglement entropy is a natural computable in nonrelativistic quantum
eld theories
that are used to describe condensed matter systems. As for relativistic systems,
entanglement entropy can be used to classify di erent phases of the system, and the behaviour of the
entanglement entropy as one scales up or down the entangling region size can elucidate the
UV and IR behaviour of the theory. Holography provides a large class of duals to strongly
interacting nonrelativistic systems, and it is interesting to explore how entanglement
entropy in these systems di ers from that in perturbative realisations of nonrelativistic
quantum
eld theories. In this section we will explore Schrodinger geometries, which indeed
exhibit an interesting transition from UV to IR behaviour. It would be interesting to
explore renormalized entanglement entropy in Lifshitz geometries also, but nontrivial
behaviour in this case requires time dependent setups  minimal surfaces at constant time
in Lifshitz behave precisely as those in AdS. We therefore postpone exploration of (time
dependent) renormalized entanglement entropy in Lifshitz for future work.
Schrodinger metrics in (p + 3) dimensions can be written as [26]
2
1
ds2 =
r2z (dx+)2 +
r2 dr2 + dx+dx
+ dxidxi
where the index i runs over p directions. The light cone coordinates can be rewritten as
x
= (y
t)
The metric can be supported by (real) massive gauge elds provided that b2 > 0 for z < 1
and b2 < 0 for z > 1.
In both cases the dual eld theory can be understood as a deformation of the CFT
by an operator that breaks the relativistic symmetry but respects nonrelativistic scaling
invariance i.e. the dual theory has the form [27]
ICF T +
Z
dp+2x jbjO
+
where the operator O
is a vector (or tensor) that picks out the x+ direction. The
deformation is relevant (dimension less than (p + 2)) with respect to the original conformal
symmetry for z < 1 and irrelevant for z > 1.
Let us consider the case of z < 1, so that the theory remains UV conformal; this case
was explored in detail in [28]. We can then specify a spacelike entangling region in the dual
eld theory, at constant t, de ned by y(xi). (For z > 1, the situation is more
complicated as surfaces of constant t are not spacelike at in nity and we will not discuss
this case further here.)
We can illustrate the behaviour of the entangling surfaces by two cases: a slab of width
L in the y direction and a slab of width L along one of the xi directions. In the latter case
the entangling surface in the bulk is described by w(r) (where w is the direction transverse
to the entangling region) and the entangling functional is
S1 =
RyVp 1 Z
2G
1
r(p+1)
p
1 + b2r2(1 z)p1 + w0(r)2dr
where now Vp is the regulated volume of the xi directions.
Note that both entangling functionals can be expressed in the form
f (r)p1 + g(r)w0(r)2dr
tional is
S2 =
Vp Z
2G
1
r(p+1)
q
1 + (1 + b2r2(1 z))y0(r)2dr
where Ry is the regulated length of the y direction and Vp 1 is the regulated volume of the
In the other case the entangling surface is described by y(r) and the entangling
func(6.1)
(6.2)
(6.3)
(6.4)
(6.5)
(6.6)
HJEP01(28)4
for suitable choices of the overall normalisation N and the functions (f (r); g(r)). Then the
width of the entangling region is given by
where ro is the turning point of the minimal surface and
L = 2
Z 1
ro
dr
pg(r)(g(r)f (r)2
g(ro)f (ro)2)
S = N
Z 1
ro
f (r)2pg(r)
pg(r)f (r)2
g(ro)f (ro)2 dr:
When f (r) and g(r) are monomials of r, the renormalized entanglement entropy can be
calculated exactly using the AdSD result derived in section 3.
For the slab along the xi directions, the renormalized entanglement entropy interpolates
between the AdSp+3 result (for small slab widths)
HJEP01(28)4
and the following result for large slab widths
(S1)ren =
RyVp 1
0 2p
0 2p
2(p+z) + 12 1p+z
1
1
2(p+z)
A
The latter expression applies for bL1 z
1, in which case the functional is approximated by
S1 =
RyVp 1 Z
2G
b
which is precisely the functional analysed in section 3 (taking D = p + z).
In the other case, the renormalized entanglement entropy is also given by the AdSp+3
result for small slab widths while at large slab widths bL1 z
1 the relevant functional is
S2 =
Vp Z
2G
1
r(p+1)
q
1 + b2r2(1 z)y0(r)2dr:
Consider rst the special case of z = 0. By a change of variable we can express this
functional as
S2 =
bVp Z
2G
w b
1p p1 + y_(w)2dw;
where y_ = dy=dw. From the general result (3.12), we can now read o the renormalized
entanglement entropy as
bVp
0 2p
b
2p
A
(6.7)
(6.8)
(6.9)
(6.10)
(6.11)
(6.12)
(6.13)
(6.14)
which is manifestly consistent with (6.9) for b = z = 1.
Thus the renormalized entanglement entropy scales di erently for large slab widths
(such that bL1 z
1), depending on the orientation of the slab with respect to the
y direction along which the theory is deformed away from conformality. In this case the
explicit symmetry breaking is associated with a breaking of the relativistic symmetry, while
preserving nonrelativistic scale invariance, and the renormalized entanglement entropy
does not have a discontinuity in its derivative and does not saturate in the deep IR. (Note
however that the Schrodinger geometry has a null curvature singularity and thus quantum
corrections to the geometry may change the deep IR behaviour.)
7
Interpretation and comparison to QFT results
In the previous sections we have explored the renormalized entanglement entropy for slab
domains in a variety of holographic models. While the general method of area
renormalization is applicable to entangling domains of any shape, it is particularly convenient to
use slab domains for several reasons. Firstly, the equations of motion admit rst integrals,
thus simplifying the analysis. Secondly, slab entangling domains have been analysed for a
variety of quantum
eld theories in the literature. Note that the previous literature does
not compute the renormalized entanglement entropy, but typically extracts instead
As we discussed in the introduction, in any local quantum
eld theory the divergences in
the entanglement entropy are necessarily independent of the width of the slab, L, and thus
c(L) is manifestly UV
nite.
Consider now the renormalized entanglement entropy. The counterterms include
nite contributions, as illustrated in the previous section, but these
nite contributions are
independent of the width of the slab, as the counterterms are expressed in terms of local
quantities at the boundaries of the entangling region. Therefore
bVp
For 0 < z < 1, the functional (6.12) can be expressed as
and thus from (3.12)
S2 =
bVp
2G
1
zb
p1 + y_(w)2dw
pz +1 Z
dw
w pz +1
pz +1 0 2p
and thus the slope of our renormalized quantity matches the c function de ned in earlier
literature. This statement can be expressed as
Z
Sren =
c(L)dL + s0
where s0 is independent of L, but dependent on parameters of the theory. Thus the
renormalized quantity is an integrated version of the c function. (Note that the c(L) is
de ned in various di erent ways in the literature. For example, [3] use a de nition of the
entropic c function that incorporates factors of L.)
Next let us consider the UV and IR behaviour of the renormalized entanglement
entropy. The renormalized entanglement entropy measures the residual entanglement
between the entangling region and its complement, after subtracting the divergent
contributions arising from entanglement at the boundary. In the ground state of a conformal eld
theory, correlation functions are characterised by power law behaviour and thus it would
be reasonable to expect that the residual entanglement scales inversely with the width of
the slab entangling region (and extensively with the length of the slab region).
This heuristic argument is in agreement with the explicit holographic result (3.12). For
a slab region in the ground state of a conformal eld theory, the L independent contribution
s0 in (7.3) is necessarily zero, as there is no dimensionless ratio that is independent of L.
The renormalized entanglement entropy is thus determined entirely by the c function, with
the positivity of the latter implying the negativity of the former. For nonconformal branes,
the entanglement entropy is controlled by the conformal structure in (d
2
) dimensions,
and therefore similar arguments apply.
Suppose that in the IR of the theory correlation functions fall o exponentially with
characteristic mass scale ; entanglement is thus signi cant only on length scales of order
1 from the entangling region boundaries. If the width of the entangling region L is
much greater than this length scale, then we would expect the renormalized entanglement
entropy to saturate to a value that is independent of L. On dimensional grounds this
residual entanglement entropy must then scale for a ddimensional theory as Vd 2 d 2 for
a slab region of area Vd 2. Thus there is a nonvanishing constant term in (7.3) which
would not be seen in the c function (which is in such cases zero for large L).
This behaviour can be seen in a number of explicit QFT calculations. In [29] the
entanglement entropy for massive scalar elds in various dimensions was computed, and
expressed in term of the derivative of the entanglement entropy with respect to the mass
. The latter is sensitive to the contributions to the renormalized entanglement entropy
that are independent of L, and hence are lost from the c function. For example, for d = 3,
it was shown that
V1
24
with V1
1 the regulated length of the slab region. Integrating this expression results
in a nite contribution to the renormalized entanglement entropy
in agreement with the above arguments.
In d = 4 the analogous expression is
S
S
(7.4)
(7.5)
(7.6)
Sren =
V1 ;
12
V2 2
48
:
These terms arise from logarithmic divergences in the regulated entanglement entropy
V2
48
S =
2 :
Whenever there is a logarithmic divergence, the renormalized entanglement entropy has
scheme dependence, corresponding to the choice of nite counterterms [6].
Such logarithmic divergences occur in particular for CFTs in d dimensions deformed
by operators of dimension
= d=2 + 1 [30, 31]. The logarithmic divergences are removed
by counterterms of the form
in holographic realisations, where
is the scalar eld dual to the deforming operator. By
rescaling
! e
this counterterm will change to
(7.7)
(7.9)
(7.10)
(7.8)
d 2xp
2 log
d
d 2xp
2(log + )
with the latter term being nite (due to the operator dimension). In particular, using the
leading asymptotic behaviour for the scalar eld, the latter term contributes a term
Vd 2 s2
to the renormalized entanglement entropy, where s is the operator source. Therefore, the
renormalized entanglement entropy depends explicitly on the choice of the coe cient of the
nite term
. (More generally, operators of dimension
= d(1
1=2n)+1=n are associated
with logarithmic divergences [6] and hence lead to nite terms in the entanglement entropy
behaving as Vd 2 s2n.)
In supersymmetric theories, the ambiguity can be xed by requiring that the
renormalization scheme for the partition function respects supersymmetry and then using the
replica trick to derive the counterterms for the entanglement entropy. The renormalization
scheme for GPPZ, which indeed corresponds to a CFT deformed by a supersymmetric
operator of dimension
= d=2 + 1, was constructed to respect supersymmetry [23, 24]. It is
thus perhaps unsurprising that the supersymmetric scheme implies that the renormalized
entanglement entropy in this case vanishes in the deep IR.
The discontinuity in the derivative of the entanglement entropy with respect to L
in a holographic con ning theory was rst described in [1]. In the examples of explicit
and spontaneous symmetry discussed here the renormalized entanglement entropy always
saturates in the IR, and there is a discontinuity in the derivative of the entanglement
entropy at the critical value of L, at which the dominant entangling surface becomes
disconnected. Note however that the slope of the derivative can be small close to the
transition point, as in one of our Coulomb branch examples, and one thus needs to ensure
that the numerical resolution is su cient to capture the discontinuity in the derivative.
In addition to calculations of the entanglement entropy in free eld theories, various
calculations of the entanglement entropy for slab regions have been carried out in lattice
gauge theories. In [
32
] the entanglement entropy for a slab of width L in a fourdimensional
SU(2) gauge theory was studied numerically. The results of this study are in agreement
with the behaviour found here. The derivative of the entanglement entropy with respect
to L has a discontinuity at a critical value, as found in holographic con ning theories in [1]
and discussed above, and it was also observed that there are
nite contributions to the
entanglement entropy which scale as 1=L2 for small width entangling regions.
While [
32
] did not extract the renormalized entanglement entropy, their results imply
that the renormalized entanglement entropy would scale as 1=L2 for small width entangling
regions. The SU(2) gauge theory is asymptotically free and thus one would expect the
renormalized entanglement entropy for small regions to be captured by free gluons, which
indeed scales in accordance with the conformal result discussed earlier in the paper. Note
that the residual nite contributions at large L were not computed in [
32
].
A more recent lattice simulation [
33
] studied entanglement entropy for slab regions in
SU(3) gauge theory in four dimensions. The generic features are similar to those found
in the SU(2) theory (free at small distances, c(L) goes to zero at
nite L), although the
detailed features near the critical length di er between SU(2) and SU(3). In particular, c(L)
seems to go smoothly to zero at the critical length, and therefore there is no discontinuity in
the derivative of the entanglement entropy with respect to L. As in [
32
], only the vanishing
of the derivative of the renormalized entanglement entropy for large L was shown; the
residual nite entanglement entropy was not computed.
8
Conclusions and outlook
In this paper we have explored renormalized entanglement entropy for slab domains, for
a variety of di erent holographic theories. We have shown that the renormalized
entanglement entropy captures not just the features of the previously discussed entangling c
function, but also the deep IR behaviour of symmetry breaking theories (where the c
function vanishes). It would be interesting to analyse the properties of renormalized
entanglement entropy for other common entangling regions, such as spheres and hypercubes.
Note however that the latter are considerably more complicated to compute
holographically: the equations of motion for the minimal surfaces do not admit rst integrals and the
vertices of hypercubes are generally associated with additional logarithmic counterterms
in the entanglement entropy.
The examples discussed in this paper indicate the existence of general bounds on the
renormalized entanglement entropy: Sren
0 with Sren ! 0 for supersymmetric RG
ows
triggered by operator deformations. It would be interesting to develop proofs of these
bounds in future work. Related bounds were discussed in [34], although the functional
analysed in [34] is not identical to the renormalized entanglement entropy considered here.
Note that there are heuristic arguments why Sren
0. For CFTs in odd dimensions,
following [35], the renormalized entanglement entropy for spherical regions is related to
the partition function on a sphere, and the negativity of the renormalized entanglement
entropy is thus related to the conjectured positivity of the F quantity [36]. (Away from the
xed points, along the RG
ow, the relationship between the F quantity (the free energy
on the sphere) and the renormalized entanglement entropy is more complicated than the
relation in [35] but nonetheless in all explicit examples positivity of F indeed maps to
negativity of the renormalized entanglement entropy for a disk region.)
More generally, the renormalized entanglement entropy coincides with minus the
(renormalized) Euclidean action for a D(d
1)brane with no worldvolume gauge
elds
and no ChernSimons couplings to background
uxes i.e. the latter is also a minimal
surface. The (renormalized) Euclidean action is positive semide nite for stable Dbrane
embeddings, and vanishes for supersymmetric Dbrane embeddings. This heuristic argument
suggests that the renormalized entanglement entropy should be negative semide nite but
does not however explain why the renormalized entanglement entropy is zero in the IR for
supersymmetric operator driven
ows but not for supersymmetric VEV driven ows.
Holography allows us to explore entanglement entropy for a wide variety of strongly
coupled quantum
eld theories. In this work we have extracted from existing perturbative
and lattice results the behaviour of the renormalized entanglement entropy for slabs but it
would clearly be interesting to explore renormalized entanglement entropy directly within
perturbative quantum
eld theory, using varied renormalization methods. The replica trick
allows us to derive the counterterms for the entanglement entropy but it would be useful to
understand the role of these counterterms in computations of renormalized entanglement
entropy via twist eld correlators.
There has been considerable progress in understanding the computation of
entanglement entropy in lattice gauge theories, see for example [32, 33, 37{39], and it would be
interesting to explore how the continuum limit of such computations can be matched with
our de nition of renormalized entanglement entropy.
More generally, one would hope that it may become possible to calculate entanglement
entropy for certain supersymmetric theories on the lattice in the near future  see for
example [40] for recent progress on simulating N = 4 SYM. We can rewrite the holographic
result (3.12) for the renormalized entanglement entropy for a slab in N = 4 SYM as
Conformal invariance implies that the renormalized entanglement entropy has a leading
behaviour at large N
Sren
0:114
Sren =
f (gY2MN )
N 2Vy
L2 :
N 2Vy
L2
(8.1)
(8.2)
where f (gY2MN ) is a positive function of the 't Hooft coupling; it is this function that one
would like to compute perturbatively using lattice simulations. One can estimate the free
eld value of this function by summing contributions from the six real scalars, four Weyl
fermions (equivalent to two Dirac fermions) and the gauge eld of N = 4 SYM. Estimating
the gauge eld contributions by scaling the recent SU(3) result of [
33
] and taking the other
contributions from [3, 41] we obtain f
0:05 at zero coupling. This suggests that the
magnitude of f increases with the 't Hooft coupling, as one might expect.
Finally, we would like to turn to issues of measurability. Throughout this paper we
have focussed on the de nition and calculation of renormalized entanglement entropy in
a UV complete quantum
eld theory. We believe that this is an important computable:
our systematic renormalization procedure makes it clear when the renormalized quantity
is scheme independent and thus when nite residual terms are meaningful. The systematic
renormalization also allows us to compare di erent phases.
One may however be interested in a system for which only a low energy e ective theory
description is known; this e ective eld theory may not be renormalizable. In such a
context, one would rst calculate the regulated entanglement entropy in terms of the UV cuto
for the system. If the e ective eld theory description is associated with a renormalizable
eld theory, one could follow the procedures of this paper to de ne renormalized
entanglement entropy. (This will indeed be the case if the IR theory is a CFT.) If however
the e ective eld theory is not renormalizable, one will inevitably need to retain the cuto
dependence in the entanglement entropy and work with the regulated quantity. In the latter
context, one will not be able to extract in a meaningful way nite contributions to the
entanglement entropy. Thus the renormalized entanglement entropy, as with other renormalized
QFT quantities, is applicable to UV complete renormalizable quantum
eld theories.
Acknowledgments
We would like to thank Antonio Rago for useful comments regarding lattice calculations of
entanglement entropy. This work was supported by the Science and Technology Facilities
Council (Consolidated Grant \Exploring the Limits of the Standard Model and Beyond").
We thank the Simons Center and the GGI for partial support during the completion of this
work. This project has received funding from the European Union's Horizon 2020 research
and innovation programme under the Marie SklodowskaCurie grant agreement No 690575.
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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